On sufficient conditions for the convergence of successive approximations (Q1974381)

The main result, stated in Theorem A, gives sufficient conditions for an operator that maps a Banach space into itself, to have a fixed point, such that the sequence of successive approximations converges to the fixed point, for a chosen initial approximation. Instead of the classical contraction condition in the well-known fixed point principle, a nonlinear expression does appear in the right-hand side of the contraction-type condition considered by the author. This weaker contraction-type condition ensures no longer the uniqueness of the fixed point. Two examples are given finally to show how to apply the obtained results to concrete equations.